Neutron Capture Cross Section of Si-28. Si-29 and Si-30

The MACS of the three Si isotopes has been calculated using the cross section data from ENDF/B-VIII.0 nuclear data library. For Si-30, the first resonance at 2.235 keV has been removed as the presence of this resonance has been questioned and its non-existence confirmed in preliminary data taken with a natural-Si sample measured at n_TOF in 2021, during the commissioning.

imgd44b02adb93c3882b379fda1a570a81c.png
Figure 1: Neutron capture cross sections of the three stable silicon isotopes taken from the ENDF/B-VIII.0 nuclear data library.

MACS
img847ec8c3b81949c942e06389a3da4371.png
Figure 2: MACS-30 for the three stable silicon isotopes taken from the ENDF/B-VIII.0 nuclear data library.


The MACS up to kT=100 keV is completely determined by the resolved resonance region.

Si-28
imgc93eedf4d6d8356a04ce4eb8e4cc5298.gif

Si-29
img5de493fe932a47de360db4a7c44c3c99.gif

Si-30
img3e9e2d4b1658bdc194a071c5a3a1209f.gif

Si-30 DRC calculation

DRC calculations have been performed using the bound-state wave-functions from Woods-Saxon mean field derived from experimental data of low-lying in Si-31. The basic information is reported here below.

# Total number of states                       
            9
#                                       
#    Bind (MeV)    J   Par     C2S     Ncomp Ncol   Ex [MeV]   
       6.5874     1.5   +1    0.7030      1    1     0.0000    
       5.8354     0.5   +1    0.2530      1    1     0.7520    
       4.8925     2.5   +1    0.0170      1    1     1.6949    
       4.2705     1.5   +1    0.0400      1    1     2.3169    
       3.7993     2.5   +1    0.0430      1    1     2.7881    
       3.4543     3.5   -1    0.5950      1    1     3.1331    
       3.0545     1.5   -1    0.4030      1    1     3.5329    
       1.8704     0.5   +1    0.1100      1    1     4.7170    
       1.3104     0.5   -1    0.4440      1    1     5.2770    

Two sets of DRC calculations have been performed. The first, adopts a neutron-nucleus mean-field interaction potential of Woods-Saxon shape, with standard geometrical parameters r0=1.236 fm, d=0.62 fm and Vso=7.0 MeV. The potential well depth, V0=54.4 MeV, has been adjusted to reproduce the thermal neutron capture cross section of 107 ± 2 mb. The DRC component of the thermal capture must add up to 90 mb, as the tails of the s-wave resonances at thermal energy is of 17 mb, using the ENDF/B-VIII.0 resonance parameters. Using the same mean-field potential, a quite large p-wave DRC component raises up, in the 10-100 keV neutron energy region as appears in the figure below.

However, in the case of Si-30, the DRC p-wave component is very sensitive to the mean-field interaction potential. This is due to the presence of the 2p single particle shape-resonance (split into the 2p3/2 and 2p3/2 components), located close to the neutron binding of Si-31. The scattering wave function, describing the initial state in the DRC scheme, is strongly affected by the interaction potential.

Because of this situation, a second set of calculations using a simple hard-sphere interaction potential (an infinitely deep potential of radius R=3.84 fm) has been considered as well. The s-wave component of the DRC remains essentially the same as that obtained using a Woods-Saxon mean-field potential, while the p-wave DRC is much smaller in this case.

img4ff449adcbba5d76c7a9d3c9ad38d539.png
Figure 6: Neutron capture cross section of Si-30 as in ENDF/B-VIII.0. The two different sets of p-wave DRC calculations are shown (see text for details).

img3571a1b4b8c32752defdbb7e81570242.png
Figure 7: BW indicates the neutron capture cross section as calculated from the resolved resonances present in the ENDF/B-VIII.0 evaluation. Only positive-energy resonances are included, while the 1/v component included is calculated from the s-wave DRC model. The two different sets of p-wave DRC calculations are shown (see text for details).

The extreme sensitivity of the DRC on the mean-field interaction potential can be illustrated by calculating the MACS-30 (maxwellian averaged capture cross section at kT = 30 keV), and varying the interaction potential well-depth. This is shown in the figure below.

imgba284bcd61ccccfa326382595c9cbd1d.png
Figure 8: p-wave MACS-30 for Si-30. The extremely large sensitivity to the mean-field interacting potential is shown. The splitting of the 2p single-particle orbit is also apparent. The MACS-30 value obtained for a potential strength of 54.4 MeV is show as the interception with a vertical bar, while the value for a hard-sphere potential (independent, of course, on the potential strength) is shown by the horizontal line.



The conclusion of this analysis is that, the DRC component in the Si-30 neutron induced reaction cannot be predicted by model calculations, in particular for the p-wave component. An accurate measurement (challenging!)in the keV neutron energy region, in between resonances, could help to solve this issue.

The p-wave contribution to the MACS of Si-30, estimated from DRC calculations are shown in the figure below.

imgbfc5c1bb9b0d4af4f0b6a9f8c69dc8a4.png
Figure 9: MACS results for all three silicon isotopes. The p-wave contribution from DRC is below 1 mb for the full temperature range. The results of the calculations from the hard-sphere mean-field interaction is completely negligible (below 1 μb for the full range).

Si-30 by activation

Estimates of the activity of the n+Si-30 -> P-31 (β-, 157.24 min) is done assuming the neutron beams of EAR2 and NEAR. Assuming a 1 g sample of Si-30

#EAR2 (latest sumulation)                     #NEAR station (M81 configuration)       
#neutron flux [n/cm2/pp] : 0.20072E+07        #neutron flux [n/cm2/pp] : 0.39600E+09
#neutron flux  [n/cm2/s] : 0.33188E+06        #neutron flux  [n/cm2/s] : 0.65476E+08
#neutron flux  [n/b/min] : 0.19913E-16        #neutron flux  [n/b/min] : 0.39286E-14
#neutron flux      [n/s] : 0.10426E+07        #neutron flux      [n/s] : 0.20570E+09
#pulse/s                 : 0.16534E+00        #pulse/s                 : 0.16534E+00
#transmission            : 0.10000E+01        #transmission            : 0.10000E+01
#sample area       [cm2] : 0.31416E+01        #sample area       [cm2] : 0.31416E+01
#SACS                [b] : 0.15290E-01        #SACS                [b] : 0.29400E-02 


img17698a9a4fe178dac85ec505eee2bcbb.png

NOTE: It is to be noted, that the 1.266 MeV γ-ray after the decay Si-31 → P-31, with t1/2 = 157.24 min, is emitted with only 0.05% intensity.

References

[1] Atlas of Neutron Resonances
Resonance Properties and Thermal Cross Sections Z = 160
Sixth Edition, Volume 1
S.F. Mughabghab
National Nuclear Data Center Brookhaven National Laboratory Upton, NY 11973-5000

-- AlbertoMengoni - 2022-12-10

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